Tài liệu hướng dẫn tự học môn Giải tích 12 - Ứng dụng của đạo hàm để khảo sát hàm số - Nguyễn Văn Tặng
7. Caùc khaùi nieäm lieân quan ñeán haøm soá:
Haøm soá cho bôûi bieåu thöùc ñöôïc kí hieäu y = f(x) vôùi f(x) laø moät bieåu thöùc chöùa bieán x.
• Taäp xaùc ñònh cuûa haøm soá: D = {x R f(x) coù nghóa}.
• Giaù trò cuûa haøm soá y = f(x) taïi x0 laø y0 = f(x0).
Ví duï 1: Giaù trò cuûa haøm soá y = x2 + 1 taïi x0 = 2 laø 5
Ví duï 2: Cho haøm soá y = f(x) = (1)
a) Tính f(2), f(-1);
b) Tính giaù trò cuûa haøm soá taïi x = -2;
c) Tìm toïa ñoä ñieåm M coù hoaønh ñoä x = 0 treân ñoà thò haøm soá (1);
d) Tìm treân ñoà thò haøm soá (1) nhöõng ñieåm coù tung ñoä baèng 0.
Ví duï 3: Tìm taäp xaùc ñònh cuûa caùc haøm soá sau:
a) y = – 2 + 3; b) y = ;
c) y = ; d) y = ;
e) y = ; f) y = .
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bieán thieân a > 1: haøm soá luoân ñoàng bieán; 0 < a < 1: haøm soá luoân nghòch bieán. Tieäm caän truïc Ox laø tieäm caän ngang. Ñoà thò ñi qua caùc ñieåm (0; 1) vaø (1; a), naèm phía treân truïc hoaønh (y = ax > 0, "x Î R) II- HAØM SOÁ LOÂGARIT: 1) Ñònh nghóa: Cho soá thöïc döông a khaùc 1. Haøm soá y = logax ñöôïc goïi laø haøm soá loâgarit cô soá a. 2) Ñaïo haøm cuûa haøm soá loâgarit: Ñònh lí 3: Haøm soá y = logax (a > 0, a ¹ 1) coù ñaïo haøm taïi moïi x > 0 vaø * Chuù yù: Ñaëc bieät . Ñoái vôùi haøm soá hôïp y = ln[u(x)] thì Ñoái vôùi haøm soá hôïp y = logau(x), ta coù: Ví duï: = ................................................................. 3) Khaûo saùt haøm soá loâgarit y = logax (a > 0, a ¹ 1) a > 1 0 < a < 1 Toùm taét caùc tính chaát cuûa haøm soá muõ y = ax (a > 0, a ¹ 1) Taäp xaùc ñònh D = (0; +¥). Ñaïo haøm y' =. Chieàu bieán thieân a > 1: haøm soá luoân ñoàng bieán; 0 < a < 1: haøm soá luoân nghòch bieán. Tieäm caän truïc Oy laø tieäm caän ñöùng. Ñoà thò ñi qua caùc ñieåm (1; 0) vaø (a; 1), naèm phía beân phaûi truïc tung. * Nhaän xeùt: Ñoà thò haøm soá y = ax vaø y = logax (a > 0, a ¹ 1) ñoái xöùng nhau qua ñöôøng thaúng y = x. Ví duï: Veõ ñoà thò hai haøm soá treân cuøng moät heä truïc toïa ñoä: a) y = 4x vaø y = log4x; b) y = vaø y = . Giaûi: ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... Baûng ñaïo haøm cuûa caùc haøm soá luõy thöøa, muõ vaø loâgarit Haøm soá sô caáp Haøm hôïp (u = u(x)) Haøm soá sô caáp Haøm hôïp (u = u(x)) (xa)' = axa - 1 (ua)' = aua - 1.u' (ex)' = ex (ax)' = axlna (eu)' = eu.u' (au)' = aulna.u' & Ghi chuù: ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................... BAØI TAÄP REØN LUYEÄN 1. Baøi taäp cô baûn: Baøi 1: Tính ñaïo haøm cuûa caùc haøm soá sau: a) y = 5x2 - 2xcosx; b) y = . Baøi 2: Tìm taäp xaùc ñònh cuûa caùc haøm soá sau: a) y = log2(5 - 2x); b) y = log3(x2 - 2x); c) y = ; d) y = . Baøi 3: Tính ñaïo haøm cuûa caùc haøm soá sau: a) y = 3x2 - lnx + 4sinx; b) y = log(x2 + x + 1); c) y = . 2. Baøi taäp naâng cao: Baøi 1: Söû duïng tính chaát ñoàng bieán, nghòch bieán cuûa haøm soá muõ, haõy so saùnh moãi caëp soá sau: a) (1,7)3 vaø 1; b) (0,3)2 vaø 1; c) (3,2)1,5 vaø (3,2)1,6; d) (0,2)-3 vaø (0,2)-2; e) vaø; d) 6p vaø 63,14. Baøi 2: Haõy so saùnh x vôùi soá 1, bieát raèng: a) log3x = -0,3; b) ; c) log2x = 1,3; d) = -1,1. CAÂU HOÛI CHUAÅN BÒ BAØI ............................................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................................................. §5. PHÖÔNG TRÌNH MUÕ VAØ PHÖÔNG TRÌNH LOÂGARIT I- PHÖÔNG TRÌNH MUÕ: 1) Phöông trình muõ cô baûn: Daïng: ax = b (a > 0, a ¹ 1) Ví duï: · Vôùi b > 0 ta coù: ax = b Û x = logab. 2x = 3 Û ................................................ · Vôùi b £ 0 ta coù: ax = b Û x Î Æ. 2x = -3 Û ............................................... 2) Caùch giaûi moät soá phöông trình muõ ñôn giaûn: a/ Ñöa veà cuøng cô soá: aA(x) = aB(x) Û A(x) = B(x) Ví duï: Giaûi phöông trình (1,5)5x - 7 = ()x + 1. Giaûi: ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... ................................................................................................................... b/ Ñaët aån phuï: Ví duï: Giaûi caùc phöông trình sau: a) 9x - 4.3x - 45 = 0; b) 6.9x – 13.6x + 6.4x = 0. Giaûi: ...................
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